Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 647

Table 23.6 Representation of the clothoidal polynomial y = j =0

x 0 ) j case study: κ 0 = 1
5000

c j ( x − ,κ 0 =

10 − 6 m − 2 ,α 0 =45 ◦ ,x 0 = y 0 =0

Coecients Formulae Value for case study

c 0 y 0 0

c 1 tan α 0 1

1
2

sec 3 α 0 κ 0 2 . 8284 ∗ 10 − 4

c 2

sec 4 α 0 3 κ 0 tan α 0 + κ 0

1
6

7 . 4667 ∗ 10 − 7

c 3

sec 5 α 0 κ 0 3 κ 0 1+5 tan 2 α 0 +10 κ 0 tan α 0

1
24

5 . 053 ∗ 10 − 10

c 4

sec 6 α 0 [15 κ 0 tan α 0 3+7 tan 2 α 0

1
120

κ 0 κ 0 1 . 013 ∗ 10 − 12

c 5 + ∗

19 + 105 tan 2 α 0 +10 κ 0 tan α 0 ]

10 − 15 c 6 1 . 258 ∗

720 sec 7 α 0 κ 0 45 κ 0 1+14 tan 2 α 0 +21 tan 4 α 0

1

+252 κ 0 κ 0 tan α 0 2+5 tan 2 α 0
+8 κ 0 6+35 tan 2 α 0

2 . 299 ∗ 10 − 18

c 7

5040 sec 8 α 0 45 κ 0 tan α 0 35 + 210 tan 2 α 0 + 231 tan 4 α 0
+9 κ 0

1

κ 0 81 + 1218 tan 2 α 0 + 1925 tan 4 α 0
+28 κ 0 κ 0 tan 2 α 0 86 + 225 tan 2 α 0
+8 κ 0 6+35 tan 2 α 0

Table 23.7 Minimal distance mapping of a point close to the clothoid, coecients of the normal equations of
polynomial type

f 0 = − X + x 0 − Yc 1 + c 0 c 1

2 c 2 Y +2 c 0 c 2 + c 1 f 1 =1 −

f 2 = −

3 c 3 Y +3 c 0 c 3 +3 c 1 c 2

4 c 4 Y +4 c 0 c 4 +4 c 1 c 3 +2 c 2

f 3 = −

f 4 = − 5 c 5 Y +5 c 0 c 5 +5 c 1 c 4 +5 c 2 c 3

6 c 4 Y +6 c 0 c 6 +6 c 1 c 5 +6 c 2 c 4 +3 c 3

f 5 = −

f 6 = − 7 c 7 Y +7 c 0 c 7 +7 c 1 c 6 +7 c 2 c 5 +7 c 3 c 4

8 c 8 Y +8 c 0 c 8 +8 c 1 c 7 +8 c 2 c 6 +8 c 3 c 5 +4 c 4

f 7 = −

n even:
f n =

( n +1) c n +1 Y +( n +1) c 0 c n +1 +( n +1) c 1 c n +( n +1) c 2 c n− 1 + ... +
( n +1) c n/ 2 c n/ 2+1
n odd:
f n =

−

( n +1) c n +1 Y +( n +1) c 0 c n +1 +( n +1) c 1 c n +( n +1) c 2 c n− 1 + ... +
( n +1) c ( n− 1) / 2 c ( n +3) / 2 + n +1
2

−

c ( n +1) / 2